Related papers: Computing discrete Morse complexes from simplicial…
Hashing method maps similar data to binary hashcodes with smaller hamming distance, and it has received a broad attention due to its low storage cost and fast retrieval speed. However, the existing limitations make the present algorithms…
Graph based clustering is one of the major clustering methods. Most of it work in three separate steps: similarity graph construction, clustering label relaxing and label discretization with k-means. Such common practice has three…
In this paper, we study some useful properties of persistent pairs in a discrete Morse function on a simplicial complex $K$. In case of $\dim K=1$ (i.e., a graph), by using the properties, we characterize strongly connectedness of critical…
Computation of the simplicial complexes of a large point cloud often relies on extracting a sample, to reduce the associated computational burden. The study considers sampling critical points of a Morse function associated to a point cloud,…
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by…
In this work, we introduce a combinatorial-geometric model for the space of discrete Morse functions on any CW complex $X$. We relate this version of a space of discrete Morse functions to the space of cellular filtrations of $X$ and…
We study perfect discrete Morse functions on closed oriented n-dimensional manifolds. We show how to compose such functions on connected sums of closed oriented manifolds and how to decompose on connected sums of closed oriented surfaces.
We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. Finite ordered sets also serve as the codomain of our…
In this paper, we study a class of discrete Morse functions, coming from Discrete Morse Theory, that are equivalent to a class of simplicial stacks, coming from Mathematical Morphology. We show that, as in Discrete Morse Theory, we can see…
This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…
We study transformations between discrete Morse functions on a finite simplicial complex via birth-death transitions--elementary chain maps between discrete Morse complexes that either create or cancel pairs of critical simplices. We prove…
We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric…
We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and…
A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined…
Geometric graphs form an important family of hidden structures behind data. In this paper, we develop an efficient and robust algorithm to infer a graph skeleton of a high-dimensional point cloud dataset (PCD). Previously, there has been…
Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the stratified Morse trajectories in a small…
A wide range of optimization problems arising in machine learning can be solved by gradient descent algorithms, and a central question in this area is how to efficiently compress a large-scale dataset so as to reduce the computational…
Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices…